Theorem spw 2034

Description: Weak version of the specialization scheme sp 2184. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2184 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2184 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2136 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2184 are spfw 2033 (minimal distinct variable requirements), spnfw 1979 (when 𝑥 is not free in ¬ 𝜑), spvw 1981 (when 𝑥 does not appear in 𝜑), sptruw 1806 (when 𝜑 is true), spfalw 1980 (when 𝜑 is false), and spvv 1988 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1910	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1910	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1910	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2033	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  hba1w  2048  19.8aw  2051  exexw  2052  spaev  2053  ax12w  2134  rspw  3223  reldisj  4433  ralidmw  4488  dtruALT2  5345  dtruOLD  5421  bj-ssblem1  36677  bj-ax12w  36700  eu6w  42666